Math 230

Test II

Bellamy & Sullivan

Fall 1998

Problems count 25 points each; problem 5 is worth 10 quiz points.

1. A fair coin is first tossed once. If the result of this first toss is a head it is tossed once again, while if the result of the first toss is a tail, it is tossed two more times.
   1. Find the expected number of heads obtained.
   2. Given that the number of heads obtained was 2, find the conditional probability that the first toss was a head.

2. An urn contains three red balls and four green balls. Three balls are drawn randomly, one after another, without replacement.
   1. Find the probability that at least two of the 3 balls drawn were green.
   2. Given that the third ball drawn was red, find the conditional probability that the first two were both green.
   3. Is the result in b) larger or smaller than the (unconditional) probability that the first two were both green?

3. Let X denote the number of heads in three tosses of a coin which is bent and weighted so that the probability of getting a head on any toss is . Find
   1. The expected number of heads.

   2. The variance of the number of heads.
   3. The standard deviation of the number of heads.
   4. The expected value of 4X + 5.

4. Mike and Bill are riding bicycles on a trip together. However, Bill rides six miles per hour faster than Mike, so Mike starts the trip two hours sooner. Bill catches up with Mike after he (Bill) has ridden for five hours. How fast is each riding, and how far have they traveled when they meet? Set this up as a system of linear equations in two unknowns. Then solve it and interpret your result, answering the questions posed.

5. You pay one dollar and then roll 5 dice. You get back a dollar for each die which yields a 3. Let W denote your net winnings on this bet. Find the expected value of W, and interpret the result.